In this chapter we introduce the concepts of dynamic stochastic models and rational expectations. Dynamic stochastic models, and an appropriate expectations hypothesis are indispensable if one were to model conditions in which there is uncertainty about the future.

Unlike the deterministic models with perfect foresight we have used so far, in which there was no uncertainty about the future, a more realistic treatment of dynamic macroeconomics requires the modelling of uncertainty and uncertain expectations about the future. In this chapter we introduce the basic concepts and discuss the prop- erties and alternative solution methods for such dynamic stochastic models under rational expectations.

Dynamic stochastic models can be contrasted with deterministic models. A dynamic deterministic model is specified by a set of differential or difference equations that describe exactly how the system will evolve over time. There is no uncertainty. The two period models of Chapter 2, and the infinite horizon growth models of Chapters 3 to 9 are examples of such deterministic models. A stochastic model describes the interactions of random variables, while a dynamic stochastic model describes the dynamic interactions between random variables and stochastic processes. If a dynamic stochastic model is run several times, it will not give identical results. Different runs of a dynamic stochastic model are different realizations of a stochastic process and imply different results. Thus, stochastic models embody uncertainty. Instead of describing a process which can only evolve in one way, as in the case of solutions of deterministic systems of ordinary differential or difference equations, in a dynamic stochastic model, there is inherent indeterminacy. Even if the initial conditions (or starting points) are known, there are several (often infinitely many) directions in which the process may
evolve over time.

Like random variables and stochastic processes themselves, dynamic stochastic models can be defined in both continuous and discrete time. In the case of discrete time, a dynamic stochastic model determines a sequence of random variables. The random variables corresponding to various time periods may be completely different, the only requirement being that these different random quantities all take values in the same space. The approach we take is to model these random variables as random functions of the time index. Although the random values of a dynamic stochastic model at different times may be independent random variables, in most commonly considered situations they exhibit complicated dynamic statistical dependence.

We start with simple models of exogenous stochastic processes, in order to illustrate the concepts of both dynamic stochastic models and rational expectations, and subsequently examine more general first and second order linear economic models with one endogenous and one exogenous variable. We apply these methods to two simple economic models, one from finance and one from monetary economics, in order to illustrate the solution methods, which, in subsequent chapters, are applied extensively to dynamic stochastic models of consumption, investment, money demand and aggregate fluctuations.

In the final part of the chapter we briefly discuss the conditions for unique solutions of multivariate rational expectations models and their alternative solution methods.

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